Assumption #1: You have one dependent variable that is measured at the continuous level.
Assumption #2: You have one independent variable that consists of three or more categorical, independent groups.
Assumption #3: You should have independence of observations, which means that there is no relationship between the observations in each group of the independent variable or among the groups themselves.
Assumption #4: There should be no significant outliers in the three or more groups of your independent variable in terms of the dependent variable.
Assumption #5: Your dependent variable should be approximately normally distributed for each group of the independent variable.
Assumption #6. You have homogeneity of variances (i.e., the variance of the dependent variable is equal in each group of your independent variable).
H₀: The mean plant weight is the same across all groups
H₁:At least one group’s average plant weight differs from the others.
A research wants to determine whether different treatments have an impact on plant growth. Specifically, they want to know if there is a statistically significant difference in the average weight of plants between the three treatment groups. Thus, the research would like to ask the following question: is there a significant difference in the mean weight of plants across the different treatment groups, namely: ctrl, trt1, and trt2?
summary(PlantGrowth)## weight group
## Min. :3.590 ctrl:10
## 1st Qu.:4.550 trt1:10
## Median :5.155 trt2:10
## Mean :5.073
## 3rd Qu.:5.530
## Max. :6.310str(PlantGrowth)## 'data.frame': 30 obs. of 2 variables:
## $ weight: num 4.17 5.58 5.18 6.11 4.5 4.61 5.17 4.53 5.33 5.14 ...
## $ group : Factor w/ 3 levels "ctrl","trt1",..: 1 1 1 1 1 1 1 1 1 1 ...Assumption #1: You have one dependent variable that is measured at the continuous level.
Remark. The weight variable is continuous, so this assumption is met.
Assumption #2: You have one independent variable that consists of three or more categorical, independent groups.
Remark. The group variable in the dataset has three distinct levels (ctrl, trt1, and trt2), which satisfies the assumptions.
Assumption #3: You should have independence of observations, which means that there is no relationship between the observations in each group of the independent variable or among the groups themselves.
Remark. Since the dataset is from an experimental setup with separate groups, we assume that observations must also be independent.
Assumption #4: There should be no significant outliers in the three or more groups of your independent variable in terms of the dependent variable.
plot_ly(PlantGrowth, y = ~weight, color = ~group, type = "box") %>% layout(title = "Plant Growth per Group")Remark. The boxplot above shows one outlier in the tr1 group, represented by a point outside the whiskers of a box. This suggests that there is a higher value compared to the majority of the data in that group, however, while this may not severely impact the ANOVA results, its presence should be noted.
Assumption #5: Your dependent variable should be approximately normally distributed for each group of the independent variable.
If p-values from the Shapiro-Wilk test are above 0.05 for each group, the data are approximately normally distributed, and this assumption is met.
by(PlantGrowth$weight, PlantGrowth$group, shapiro.test)## PlantGrowth$group: ctrl
##
## Shapiro-Wilk normality test
##
## data: dd[x, ]
## W = 0.95668, p-value = 0.7475
##
## ------------------------------------------------------------
## PlantGrowth$group: trt1
##
## Shapiro-Wilk normality test
##
## data: dd[x, ]
## W = 0.93041, p-value = 0.4519
##
## ------------------------------------------------------------
## PlantGrowth$group: trt2
##
## Shapiro-Wilk normality test
##
## data: dd[x, ]
## W = 0.94101, p-value = 0.5643Remark. Since the p-values of all groups are greater than 0.05, we fail to reject the null hypothesis of the Shapiro-Wilk test. This suggests that the weight variable is normally distributed within each group, meeting the normality assumption for ANOVA.
Assumption #6. You have homogeneity of variances (i.e., the variance of the dependent variable is equal in each group of your independent variable).
If the Levene’s test p-value is above 0.05, the variances are equal across groups, satisfying the assumption.
leveneTest(weight ~ group, data = PlantGrowth)## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 2 1.1192 0.3412
## 27Remark. Since the p-value of 3.412 is greater than 0.05, we fail to reject the null hypothesis of equal variances. This indicates that the assumption of homogenity of variances is met, allowing us to proceed with ANOVA.
If p < 0.05, the groups are significantly different, so we proceed with a post hoc test. If p ≥ 0,05, there is no significant difference between the groups and there is no need for a post hoc test.
anova_model <- aov(weight ~ group, data = PlantGrowth)
summary(anova_model)## Df Sum Sq Mean Sq F value Pr(>F)
## group 2 3.766 1.8832 4.846 0.0159 *
## Residuals 27 10.492 0.3886
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Remark. Since the p value of 0.0159 < 0.05, there is a significant difference in weight between at least two of the treatment groups. Thus, we proceed with the post hoc test.
tukey_results <- TukeyHSD(anova_model)
tukey_results## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = weight ~ group, data = PlantGrowth)
##
## $group
## diff lwr upr p adj
## trt1-ctrl -0.371 -1.0622161 0.3202161 0.3908711
## trt2-ctrl 0.494 -0.1972161 1.1852161 0.1979960
## trt2-trt1 0.865 0.1737839 1.5562161 0.0120064Remark. There is a significant difference in weight between the trt2 and trt` groups (p=0.012), which means trt2 has a significantly higher weight than trt1. Moreover, there are no significant differences between ctrl and trt1 or ctrl and trt2, as both groups have a p value that is greater than 0.05.
This suggests that the trt2 treatment had a distinct effect on plant growth compared to trt1, but neither treatment differed significantly from the control group in terms of weight.
A one way ANOVA test was conducted to determine if there is a significant difference in plant weight across three groups subjected to different treatments: a control group and two treatment groups (treatment 1 and treatment 2). Assumptions were check before proceeding to ANOVA. Visual inspection revealed that there is one outlier, however it does not significantly affect the results; the Shapiro-Wilk test indicated that the data were normally distributed for each group as the p-value is > 0.05. Levene’s test confirmed the homogenity of variances (0.3412). The results of the ANOVA revealed a significant difference in plant weight between the groups,F(2, 27) = 4.846, p = 0.0159. This suggests that at least one treatment group differed in plant weight compared to others.
In the post hoc analysis using Tukey’s test, no statistically significant differences were found between treatment 1 and the control group (mean difference = -0.37, 95% CI [-1.06, 0.32], p = 0.391) or between treatment 2 and the control group (mean difference = 0.49, 95% CI [-0.20, 1.19], p = 0.198), but there was a statisitcal difference between the plant weight from treatment 1 to treatment 2 ((mean difference = 0.87, 95% CI [0.17, 1.56], p = 0.012).
In conclusion, the findings suggest that treatment 2 significantly increased plant weight compared to treatment 1, despite not having any significant differences between the dependent and independent groups. Thus, this suggests that treatment 2 is much more effective on plant growth compare to treatment 1.